Upper and lower bounds
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In mathematics, particularly in order theory, an upper bound or majorant<ref name=schaefer/> of a subset Template:Mvar of some preordered set Template:Math is an element of Template:Mvar that is Template:Nobr every element of Template:Mvar.<ref name="MacLane-Birkhoff" /><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Dually, a lower bound or minorant of Template:Mvar is defined to be an element of Template:Mvar that is less than or equal to every element of Template:Mvar. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized<ref name=schaefer/> (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
ExamplesEdit
For example, Template:Math is a lower bound for the set Template:Math (as a subset of the integers or of the real numbers, etc.), and so is Template:Math. On the other hand, Template:Math is not a lower bound for Template:Mvar since it is not smaller than every element in Template:Mvar. Template:Math and other numbers x such that Template:Math would be an upper bound for S.
The set Template:Math has Template:Math as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that Template:Mvar.
Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, but not both. An infinite subset of the rational numbers may or may not be bounded from below, and may or may not be bounded from above.
Every finite subset of a non-empty totally ordered set has both upper and lower bounds.
Bounds of functionsEdit
The definitions can be generalized to functions and even to sets of functions.
Given a function Template:Italics correction with domain Template:Mvar and a preordered set Template:Math as codomain, an element Template:Mvar of Template:Mvar is an upper bound of Template:Italics correction if Template:Math for each Template:Mvar in Template:Mvar. The upper bound is called sharp if equality holds for at least one value of Template:Mvar. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.
Similarly, a function Template:Mvar defined on domain Template:Mvar and having the same codomain Template:Math is an upper bound of Template:Italics correction, if Template:Math for each Template:Mvar in Template:Mvar. The function Template:Mvar is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set.
The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤.
Tight boundsEdit
An upper bound is said to be a tight upper bound, a least upper bound, or a supremum, if no smaller value is an upper bound. Similarly, a lower bound is said to be a tight lower bound, a greatest lower bound, or an infimum, if no greater value is a lower bound.
Exact upper boundsEdit
An upper bound Template:Mvar of a subset Template:Mvar of a preordered set Template:Math is said to be an exact upper bound for Template:Mvar if every element of Template:Mvar that is strictly majorized by Template:Mvar is also majorized by some element of Template:Mvar. Exact upper bounds of reduced products of linear orders play an important role in PCF theory.<ref>Template:Cite journal</ref>